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Valuations on Distributive Lattices II

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Classic Papers in Combinatorics

Part of the book series: Modern Birkhäuser Classics ((MBC))

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Abstract

We continue the study begun in part I [Arch. Math. 24, 230–239 (1973)] of the valuation ring of a finite distributive lattice. We show that it is the Mobius algebra of the set of join.irreducible elements and we derive Solomon’s formula for idempotents. We use the duality between posets and distributive lattices given in part I to derive mapping properties of Möbius algebras. From this we get theorems on extending finitely additive measures, theorems of Rota concerning Möbius functions, an identity due to Klee, a factorization theorem of Stanley and Greene, and results on the characteristic valuation of a distributive lattice.

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Author information

Authors and Affiliations

  1. Mathematics Department, University of North Carolina, Chapel Hill, North Carolina, 27514, USA

    Ladnor Geissinger

Authors
  1. Ladnor Geissinger
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Brandeis University, Waltham, MA, 02454, USA

    Ira Gessel

  2. Department of Mathematics, Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA

    Gian-Carlo Rota

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© 2009 Birkhäuser Boston, a part of Springer Science+Business Media, LLC

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Geissinger, L. (2009). Valuations on Distributive Lattices II. In: Gessel, I., Rota, GC. (eds) Classic Papers in Combinatorics. Modern Birkhäuser Classics. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4842-8_38

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  • DOI: https://doi.org/10.1007/978-0-8176-4842-8_38

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-4841-1

  • Online ISBN: 978-0-8176-4842-8

  • eBook Packages: Springer Book Archive

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